660 ^^ WALLACE ON A FUNCTIONAL EQUATION. 



and the angle (p, to a series of values of a;, from x=0 to a7=.01, the common dif- 

 ference of the values of x being .0001. The second gives the values of f{x) and 

 F {x) and ^, from x=M to x=\, the common difference being .01 ; and farther, 

 from x=\ to 07=5, the common difference being .05. In the third Table, instead 

 of a series of values of x increasing by a common difference, there are given the 

 values of x, /(x), and F (x) to a series of angles ^, increasing by a common differ- 

 ence of half a degi'ee. In this table the values of x are Nepeh's logarithms of the 

 tangents of (45° + i<^). The ordinate /(^) is the natural secant, and the arc F 

 {x) the natural tangent of that angle. These tables, I presume, are sufficient for 

 all applications of the catenary to the construction of bridges of suspension and 

 of equilibration. 



58. The second table alone gives the values of the ordinate and arc of the 

 curve to values of x, which differ by ^th of the parameter from ar = to a; = 1 ; 

 but, by the first and second tables used together, we may find the same to values 

 of X which differ by joJooth of the parameter, by the formula for /(x + /i), and 

 F (x + h) : here x expresses the tenths and hundredths of the given value of x, and 

 h the thousandths and ten thousandths. 



As an example, let the values of /(a;) and F {a^) be required to x+h^.8S27 : 

 In this case, 



x=z .83, fx = 1.36408,40133, F («) = 0.92863,47270 ; 



/( = .0027, /(/*) = 1.00000,36450 F (/«) 0.00270,00033. 

 And the formulae for calculation are 



/(z + h) =/(x) .f{h) + F (^) F (/O : F (;. + A) = F {x)f{h) +f(x) F (A). 



We may be satisfied with seven correct figures of the result, then we may neglect 

 two figures of each tabular number, and, using contracted multiplication, have 

 /(a;) /(A) =1.36468898 F (a:)/ (A) =0.92863810 



F(a;)F(/0= .00250731 /(«)F(/0= 0.00270001 



/(.8327) = 1.3671963. F (.8327) 0.9313381. 



If, instead of seven, no more decimal places are required than are given of the 

 value of X (viz. four), we may then take only five figures of the given tabular 

 numbers, and now we have 



/(a:)/ (A) =1.36468 F («)/(/*) = 0.92863 



F(a:)F(A)= .00251 /(a;) F (A) =0.00270 



/(.8327) =1.3662. /(.8327)= 0.9313 . 



From the first and second tables a more extensive one may be fonned by 

 interpolation and prolongation ; indeed it was partly with this view that the num- 

 ters have been carried on to so many places of decimals. 



